1.1.1 Goals of this lecture

• Some additional crucial details
  • Estimation
  • Model comparisons
  • Adding and centering predictors

2.1.1 Schizophrenia over time

• Schizophrenia treatment effects over the course of 7 weeks (\(N\) = 437), measured by the Inpatient Multidimensional Psychiatric Scale (IMPS)
  • id: ID variable
  • imps79: Continuous measure of schizophrenia (1 to 7)
  • imps79b: Binary measure of schizophrenia (3.5+)
  • imps79o: Ordinal measure of schizophrenia (Cuts: 2.5+, 4.5+, 5.5+)
  • tx: Placebo (0) or treatment (1)
  • week: Week of study (0, 1, 3, 6)

2.1.2 Data

2.2.1 Marginal model

Call:
geeglm(formula = imps79b ~ 1 + week, family = binomial("logit"), 
    data = schizx1, id = schizx1$id, corstr = "unstructured")

 Coefficients:
            Estimate  Std.err  Wald            Pr(>|W|)    
(Intercept)  2.59459  0.11876 477.3 <0.0000000000000002 ***
week        -0.45017  0.02767 264.7 <0.0000000000000002 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation structure = unstructured 
Estimated Scale Parameters:

            Estimate Std.err
(Intercept)   0.9646  0.1007
  Link = identity 

Estimated Correlation Parameters:
          Estimate Std.err
alpha.1:2  0.05390 0.05343
alpha.1:3 -0.02855 0.03265
alpha.1:4 -0.01890 0.03342
alpha.2:3  0.56341 0.10114
alpha.2:4  0.15242 0.06617
alpha.3:4  0.51550 0.07964
Number of clusters:   437  Maximum cluster size: 4 

2.2.2 \(ln\left(\frac{\hat{p}}{1-\hat{p}}\right) = 2.59 - 0.45 (week)\)

2.2.3 Population-averaged effects

\[ln\left(\frac{\hat{p}}{1-\hat{p}}\right) = b_0 + b_1 (week) = 2.59 - 0.45 (week)\]

• Basically treats week as a non-repeated measures predictor
  • Does not link observations from the same person together
• Odds ratio: \(e^{-0.45} = 0.64\)
  • Each week, the odds of diagnosis (imps79b) is multiplied by 0.64

2.2.4 Population-averaged effects

\[ln\left(\frac{\hat{p}}{1-\hat{p}}\right) = b_0 + b_1 (week) = 2.59 - 0.45 (week)\]

• Basically treats week as a non-repeated measures predictor
  • Does not link observations from the same person together
• Predicted probabilities

2.3.1 Conditional model

Generalized linear mixed model fit by maximum likelihood (Laplace
  Approximation) [glmerMod]
 Family: binomial  ( logit )
Formula: imps79b ~ 1 + week + (1 + week | id)
   Data: schizx1

     AIC      BIC   logLik deviance df.resid 
  1291.6   1318.4   -640.8   1281.6     1564 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-2.8446  0.0898  0.1139  0.2646  1.0822 

Random effects:
 Groups Name        Variance Std.Dev. Corr 
 id     (Intercept) 4.413    2.101         
        week        0.711    0.843    -0.13
Number of obs: 1569, groups:  id, 437

Fixed effects:
            Estimate Std. Error z value            Pr(>|z|)    
(Intercept)    4.386      0.539    8.14 0.00000000000000041 ***
week          -0.793      0.118   -6.71 0.00000000001954283 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
     (Intr)
week -0.817

2.3.2 \(ln\left(\frac{\hat{p}}{1-\hat{p}}\right) = 4.39 - 0.79 (week)\)

2.3.3 Person-specific effects

\[ln\left(\frac{\hat{p}}{1-\hat{p}}\right) = b_0 + b_1 (week) = 4.39 - 0.79 (week)\]

• Models each person’s trajectory separately
  • Averages intercepts to get average intercept, slopes to get average slope
• Odds ratio: \(e^{-0.79} = 0.45\)
  • Each week, the odds of diagnosis (imps79b) is multiplied by 0.45

2.3.4 Person-specific effects

\[ln\left(\frac{\hat{p}}{1-\hat{p}}\right) = b_0 + b_1 (week) = 4.39 - 0.79 (week)\]

• Models each person’s trajectory separately
  • Averages intercepts to get average intercept, slopes to get average slope
• Predicted probabilities

2.4.1 Comparison

• Marginal model ignores individual variability
  • Only cares about correlations among repeated measures
• Remember the more complex conditional models we looked at last time
  • We couldn’t include a random slope in the model with tx and week
  • Implies that people don’t really vary in their slopes over time
• The models are quite similar
  • So maybe, in this case, it doesn’t matter much to ignore the individual

3.1.1 Review: Maximum likelihood estimation (MLE)

• Models (linear, logistic regression) have a likelihood function that gives the likelihood of different parameter estimates
  • Likelihood \(\approx\) probability
• “Maximum likelihood estimates” are the parameter estimates (e.g., regression coefficients, etc) that are most likely given the data
  • Uses calculus (derivatives) to find it
• Traditional MLE requires joint distributions for the model
  • Which we don’t always have for categorical outcome models

3.1.2 Figure: Maximum likelihood estimation

3.1.3 Estimation approaches

• Marginal models
  • Generalized estimating equations (GEE)
• Conditional models
  • Likelihood approximation
    • Integral approximation, numerical integration, adaptive quadrature
  • Linearization methods
    • Pseudo-likelihood, quasi-likelihood

3.1.4 Generalized estimating equations (GEE)

• Not likelihood functions
  • Type of quasi-likelihood, so no -2LL, AIC, LR tests, etc
• Only require marginal distributions
  • Not joint as for traditional MLE
• Technically GEE is the estimation method
  • But often used to refer to marginal models in general

3.1.5 Likelihood approximation

3.1.6 Linearization methods

• Tries to turn the non-linear problem into a linear problem
  • Uses Taylor series expansion
• Pseudo- or quasi-likelihood approach
  • No -2LL, AIC, LR tests, etc
• Only option with SPSS genlinmixed
  • Default with SAS glimmix (but likelihood approximation too)

3.1.7 GLMM estimation in R

• glmer() function in lme4 package
  • Default: Laplace approximation
  • Option: Adaptive Gauss-Hermite quadrature
• Other R packages that can run GLMMs have other options

4.1.1 Model comparisons

• Whether and how you can compare models depends on
  • Which models they are
    • Conditional vs marginal
  • How they were estimated
    • Likelihood method or not

4.1.2 Marginal models

• Estimated using generalized estimating equations
  • A quasi-likelihood approach
    • No LL: No AIC, no LR tests
  • QIC is a “quasi” information criteria
    • Can be used to compare nested or un-nested models
    • Similar to AIC

4.1.3 Conditional models

• Estimated with a linearization method (not preferred)
  • A quasi-likelihood approach
    • No LL: No AIC, no LR tests
    • Can use QIC (if available) similarly to AIC
• Estimated with likelihood approximation methods (preferred)
  • You get a log-likelihood and everything that comes with it: AIC, LR tests
  • Compare nested and non-nested models as usual

5.1.1 Predictors

• Marginal models
  • Predictors are predictors
  • Everything is at one level
• Conditional models
  • Predictors can be at level 1 or level 2
    • Longitudinal: Level 1 = observation, level 2 = person
    • Cross-sectional: Level 1 = person, level 2 = class, company, etc.
  • Entered into different parts of the model

5.1.2 Predictors in the model

• Two predictors: week (L1: Observation) and tx (L2: Person)

• Level 1: Within-person equation

  • \(\eta_i = \pi_{0i} + \pi_{1i}\color{red}{(week_{ij})} + e_{ij}\)

• Level 2: Between-person equation

  • \(\pi_{0i} = \beta_{00} + \beta_{01}\color{blue}{(tx_i)} + r_{0i}\)
  • \(\pi_{1i} = \beta_{10} + \beta_{11}\color{blue}{(tx_i)} + r_{1i}\)

5.1.3 Why do we care?

• Level 1 observations have both level 1 and level 2 information
  • Longitudinal: Occasion and person
  • Cross-sectional: Person and class, company, neighborhood
• If you ask me one day if I’m depressed, that gives you information about
  • How depressed I am that day (occasion, L1)
  • How depressed I generally am (person, L2)
• How can we disentangle those two kinds of information?
  • Centering

5.2.1 Centering in multi-level models

• Grand mean centering (GMC)
  • Center all observations at the grand mean of all observations
  • Doesn’t change the relationships among variables
• Centering within cluster (CWC)
  • Center each person’s observations at the mean of that person
  • Does change the relationships among variables

5.2.2 Figure: Uncentered with grand mean

5.2.3 Figure: Grand mean centered

5.2.4 Figure: Uncentered with person (cluster) means

5.2.5 Figure: Centered within cluster

5.2.6 GMC vs CWC

• Centering changes the context for the different clusters (L2: People)
  • GMC maintains mean differences between people on L1 predictor
    • What is a person like compared to other people?
  • CWC eliminates differences between people on L1 predictor
    • What are people like compared to their own mean?
• Different contexts means different interpretations for both level 1 and level 2 predictors

5.2.7 Fully unconflated model

• Just centering doesn’t fully unconflate level 1 and level 2
• When you have predictors at level 1 and you center within cluster
  • Removed the cluster-level means: L1 and L2 are still conflated
• What to do?
  • Add cluster mean of level 1 predictor back as a level 2 predictor
• Less commonly done for longitudinal
  • More common for cross-sectional

5.2.8 Fully unconflated model

• Two predictors: week (L1: Observation) and tx (L2: Person)
  • Add L1 predictor (L1pred), which is centered within cluster (person)
• Level 1: Within-person equation
  • \(\eta_i = \pi_{0i} + \pi_{1i}(week_{ij}) + \color{blue}{\pi_{2i}(L1pred_{ij} - \overline{L1pred}_{i})} + e_{ij}\)
• Level 2: Between-person equation
  • \(\pi_{0i} = \beta_{00} + \beta_{01}(tx_{i}) + \color{red}{\beta_{02}(\overline{L1pred}_{i})} + r_{0i}\)
  • \(\pi_{1i} = \beta_{10} + \beta_{11}(tx_{i})\)
  • \(\pi_{2i} = \beta_{20} + \beta_{21}(tx_{i})\)

6.1.1 Summary of this week

• Reviewed marginal and conditional models
  • Different interpretations, different numbers
• Estimation
• Model comparison
• Predictors and centering

6.1.2 Summary of this section

• Repeated measures models for categorical outcomes
  • Marginal: \(\textbf{R}\) matrix, population averaged, GEE, cluster robust
  • Conditional: \(\textbf{G}\) matrix, cluster-specific, generalized linear mixed models (GLMM)
• Additional complexities: marginal and conditional are not the same, estimation is more difficult, model comparison is more difficult

6.1.3 Next weeks

• Next week: No class, but work on final project
  • Sign up for a meeting with me if you want to chat about anything
  • Email me if you have any questions about anything
  • Last article discussion (4/9) and homework 4 (4/16)
• 2 weeks from now: No class
  • Record presentations (4/23)
  • Watch presentations (4/26)
  • Comment on presentations (4/26)
  • Final paper (4/28)